Problem Description

In this problem, you're given a string s. Your task is to find the length of the longest substring of s where each of the vowels - 'a', 'e', 'i', 'o', 'u' - appears an even number of times. Substrings are continuous parts of the original string, and in this case, any substring that satisfies the vowel condition could potentially be the longest. The goal is to figure out the maximum length possible while adhering to the even occurrence condition for vowels.

Intuition

The intuition behind the solution is to use a bitwise representation to track the count of vowels in the substring efficiently. Because we only care about even or odd occurrences, we don't need to keep a count of each vowel; we only need to know if a vowel has appeared an even or odd number of times. Thus, each vowel can be represented with 1 bit, resulting in a 5-bit integer for all five vowels (where 0 means even and 1 means odd number of occurrences).

The key insight is that if at two different points in the string, the 5-bit integer (or state) is the same, it means that we have a substring (between these two points) where all vowels have appeared an even number of times.

The solution iterates through the string and toggles the respective bit in the state when a vowel is found. If the state has been seen before, we calculate the length of the new substring ending at the current character and update the answer if it's longer than the previous maximum. The pos array keeps track of the first occurrence of each state to calculate lengths of valid substrings. We start by initializing the pos array with inf to denote that those states have not been seen yet, except for the state 0, which is initialized to -1 to handle the edge case where a valid substring starts at the beginning of the input string.

By using this bitwise state and the pos array, we can efficiently find the longest substring where all vowels appear an even number of times.

Learn more about Prefix Sum patterns.

Solution Approach

The solution utilizes a bitwise approach combined with a hash map (implemented as a list called pos in the code) to keep track of the indices at which each state occurs for the first time.

Key Components of the Solution:

• Vowels Bitmasking: Each vowel has a corresponding bit in a 5-bit integer, where the order is 'a', 'e', 'i', 'o', 'u'. For example, if only 'a' and 'i' have been seen an odd number of times, the state would be 10100 in binary, which is 20 in decimal.

• Tracking States: The state variable is a 5-bit integer representing the current state of vowels' counts (even/odd). The initial state is 0 since we start with an even count (zero occurrences) for all vowels.

• Index Memory (pos): An array called pos of size 32 (since there are 2^5 possible states due to 5 vowels) is used to remember the earliest occurrence of every state. This array is initialized with inf, except for pos[0] which is set to -1.

• Iterating Through the String: As we iterate through the string, we check each character. If it's a vowel, we flip the corresponding bit in the state using the exclusive OR (XOR) operation: state ^= 1 << j, where j is the index of the vowel in the string 'aeiou'.

• Updating the Answer: With each new character processed, we check if the current state has been encountered before:

  • If it's been seen, we calculate the length of the substring from the first occurrence of this state to the current position.
  • If the calculated length is greater than the ans (which keeps track of the maximum length seen so far), we update ans.

• First Occurrence: We also check if the current state's first occurrence needs to be updated in the pos array. If the current index is less than the stored value in pos[state], we update pos[state] with the current index.

By the end of the string traversal, ans will hold the length of the longest substring where all vowels appear an even number of times, and that's what we return as the solution.

Example Walkthrough:

Let's walk through a quick example:

s = "eleetminicoworoep"

• We start at state = 0 and pos[0] = -1 because we have an even count (zero) for all vowels at the start.
• Iterating over s, whenever we encounter a vowel, we update state. Suppose state becomes 3 after some operations; this means 'a' and 'e' have been seen an odd number of times so far.
• If state is 3 again at a later point, we know that between these two indices, 'a' and 'e' must have appeared an even number of times. Therefore, we calculate the length of this substring and check if it's the maximum.
• We continue this process until the end of the string, constantly updating ans with longer valid substrings as we find them.

The implementation is efficient with a time complexity of O(n), where n is the length of the string, because we only need a single pass through the string to compute the result, and each operation within that pass is of constant time complexity.

Example Walkthrough

Let's illustrate the solution approach using the string:

s = "aeiobcdf"

1. Initialize the pos array with length 32 to represent all possible states of vowel occurrences (2^5 for 5 vowels). Set all values to inf, except pos[0] to -1.

2. We start with state = 0 (since no vowels have been seen yet) and ans = 0 (since no substrings have been found yet).

3. As we iteratively check each character in s, we use the following steps:

   • When at index 0, s[0] = 'a': We see our first vowel 'a'. The bit for 'a' in the state is toggled from 0 to 1. Now, state = 00001.
   • At index 1, s[1] = 'e': We toggle the bit for 'e'. Now, state = 00011.
   • At index 2, s[2] = 'i': Toggling the 'i' bit: state = 00111.
   • At index 3, s[3] = 'o': Toggling 'o': state = 01111.
   • At index 4, s[4] = 'b': 'b' is not a vowel; state remains 01111.
   • At index 5, s[5] = 'c': 'c' is not a vowel; state remains 01111.
   • At index 6, s[6] = 'd': 'd' is not a vowel; state remains 01111.
   • At index 7, s[7] = 'f': 'f' is not a vowel; state remains 01111.

4. We've reached the end of the string, and throughout, each vowel has appeared exactly once, so their counts are odd. Our final state is 01111.

5. Throughout our iteration, the state has changed from 00000 to 01111. After toggling the bits whenever we encountered a vowel, we checked pos to see if the current state had been recorded before. In this instance, since no state repeated other than the initial state 0, the length of the longest valid substring is zero because we have not encountered a state twice where the state would be 00000 again (meaning that all vowels have an even count).

So, the ans remains 0 in this example; there are no substrings where each vowel occurs an even number of times.

The key takeaway is the tracking of vowel occurrences through bit manipulation, using XOR to toggle between even and odd counts, and using the pos array to store the first occurrence of a state. This approach allows for quick lookups and updates while maintaining an efficient assessment of the longest valid substring. However, in our example, there were no repeated states to determine a valid substring. In a longer string with repeated vowel occurrences, the ans would likely be greater than zero.

Solution Implementation

Time and Space Complexity

Time Complexity

The given code iterates over each character of the string s, which has a length n. It performs a constant amount of work for each character: checking if the character is a vowel, possibly flipping a bit in the state variable, and updating ans and pos[state]. Because these operations all have a constant time complexity, the overall time complexity of the function is O(n).

Space Complexity

The space complexity is determined by the fixed-size array pos, which has 32 elements (one for each possible state of the 5 vowel bits), and a few additional integer variables (state, ans, and the loop variables). Therefore, the space complexity of the algorithm is O(1) since the space required does not grow with the size of the input string s.

Learn more about how to find time and space complexity quickly using problem constraints.